Abstract

We introduce a functional type of convergence from which one can deduce central limit theorem type results, as well as local limit theorems and precise large and moderate deviations estimates. In particular this provides us with a tool which predicts the scale up to which Gaussian approximation is valid and which explains quantitatively how a breaking of symmetry occurs at this scale. On our way, we also prove Berry-Esseen type estimates. We shall illustrate the methods with various examples:

#sums of dependent random variables with applications to the subgraph counts in the Erdos-Renyi random graph model;

#examples from random combinatorial structures;

#examples from number theory;

#examples from random matrix theory.

#examples from simple statistical mechanics models.

All these examples exhibit some dependence structure. Finally, we shall try to see how these ideas can apply to some simple financial